So in this segment, we're going to be talking about correlation. Correlation measures the association between variables. To give some examples, let's start with cooking ability which we dealt with in the previous section. And we're going to relate that to age. And we already know two points here. Your roommate is young and is not a very good cook and Grandma is old and she's a great cook. And if you were to collect additional data, you would find a tendency for people who are not such great cooks. Not maybe as bad as you're roommate, but not terrific are going to be youngish and people who are better than the Average Joe are going to be relatively older. And that gives us the correlation. Which is indicated by the diagonal here. I'd like to give you some examples of correlations of various magnitudes. Correlations range between minus one and plus one. A minus one correlation means that there is a perfect correlation such that the higher you go on the x variable, on the bottom, the lower you go on the y variable. At the other end, correlation of +1.0 indicates that the higher you go on the variable x, the higher you go on the variable y. In between, you have what would be considered a very modest correlation, a correlation of 0.3, and 0.5 would be considered moderately high correlation. Notice that the 0.3 correlation is very hard to see from this so-called scatterplot, because you're scattering these dots across this figure. You hardly see if there's actually a relationship there. With correlation of 0.5, it's easier to see that there's a correlation. And with a correlation of 0.8 it's totally obvious that there's a substantial correlation. Correlations of 0.5 are considered moderate. Correlations of 0.7 or higher are considered strong. Although this varies from field to field and problem to thought problem but this is rough way to calculate degree of association. So I'm going to give you examples of some correlations that you encounter in your life. The correlation with SAT scores with college GPA is about 0.4. The correlation between the heights of mothers and the heights of their daughters is about 0.5. The correlation between the heights of individuals with their weights, what would you think that should be? Pretty high because in general taller people are heavier. But it's not a perfect relationship and in fact the correlation is about 0.70. The degree of overweight, with the degree of cardiovascular illness. That correlation is about 0.3. And the correlation between blood pressure and cardiovascular illness is about 0.3. Those are very modest correlations, but they're big enough because you care a lot about the variable of cardiovascular illness. It's enough to do something about. Or you can correlate the height of basketball players with the points scored for the season. I don't know what that would be. You can correlate the average temperature in the Rocky Mountains on a given summer day with the number of forest fires that day and so on. You can correlate almost any variable with almost any other variable. There are two basic ways of looking at correlations. One is rank order correlation. You put one variable on one side, so mother's height ranging from rank one, the shortest, to rank ten. And then, you note that the shortest mother has a daughter who really has kind of a middling height. The next shortest mother has a daughter who's very short. Here's a sort of short mother with a sort of short daughter, and so on. And when you perform the computations here, you find that the rank order correlation is about 0.54. You can also calculate correlations by looking at the actual numbers in question, so this is height in centimeters. Where, again, you see the shortest mother has a middling height, daughter, the next shortest, has a very short daughter and son on. And this also gives you a correlation which is about 0.5. The very first correlation, by the way, that we've come to know about in history is by Pearson, who invented the Pearson Product Moment Correlation which is what I just showed you an example of. And he had a thousand cases of where he'd measured fathers heights and sons heights. And this is the scatter plot you get, this is the distribution. Notice that things are very dense, many, many points right here toward the middle of both distributions. That reflects the fact that these are both normally distributed, so you're going to get the greatest density in what corresponds to the middle of the two distributions. And even though you get a substantial concentration there and substantial degree of association. You can still get the events like this very tall father as a totally average height son and this very tall son, has a totally average height father. Correlations are the way we assess reliability of measures. There's two different ways to define reliability, one is it's the degree to which a measure of a particular variable gives the same value across occasions. Or the degree to which a measure correlates with itself. So as an example, you can have the correlation between measures of height taken on different occasions and you would expect that correlation, that is to say the reliability, to be about 1.0. That's if you're measuring to a tolerance of 0.5 inch. If you're measuring to the tolerance of a micron, you're going to have terrible reliability because the person took a slightly deeper breath this time than last time and that gets a different number. The correlation between SAT math score obtained on two different occasions is about 0.85. So the SAT people, when they tell you that you're probably going to get a very similar score if you take the test a second time, they're right. On the other hand, there's lots of wiggle room there. And some people who do not so great the first time around do a lot better the second, and vice versa. Or we can look at the correlation between Dr. Jones' judgement about whether your tooth needs filling on Monday with Dr. Jones' judgement on Thursday. That is across two different occasions. You might think that that's close to 1.0 but it's not, it's 0.8. Too bad. You went to Dr. Jones on Monday and you got your tooth fill but if you'd waited until Thursday she might have made a different judgment. A second kind or reliability is the degree to which two different measures which are supposed to measure the same thing actually give the same result. So the degree to which two different measures seem to be measuring the same thing can give some examples. Correlation between IQ test A and IQ test B. If it's low, we know that at least one of them is unreliable. But in fact, the test that call themselves IQ test tend to correlate in the range of 0.8 to 0.9. They're quiet reliable. The correlation between Dentist Jones' judgement about whether a tooth needs filling and Dentist Smith's judgement Is about 0.70. They're reliable but they're not perfectly reliable by any means. Less reliable than a person is with himself or herself. Think about what you might expect the correlation to be between two reviewers ratings of solid state physics proposal submitted to the National Science Foundation. Boy, that's really hard. Science and you might think that surely, there would be very strong correlation. Actually, there isn't. The correlation is about 0.3 and the correlation for judgments of psychology proposals is even less than that. This seems surprising because we don't necessarily recognize that we're not looking at the full range of solid state physics proposals like Joe where we say whose never taken solid state physics. So propose something here. So we don't have that kind of range from perfectly terrible to terrific, what we've can find range to be reasonable to terrific. And when you have a narrower range like that, you're going to get a relatively low correlation. And this is something we'll be taking up again in the next lesson on the law of large numbers. Because you need to pay attention of the reliability of any kind of subjective measurement that you have. Even though it's a subjective measurement of something that's really very objective in some sense. Or we could see how reliable two sorority sisters are. Or a judgment, for example, of shyness. That happens to be about 0.65. So sorority sisters are pretty darn reliable for judgments of shyness. More reliable, incidentally, than they are for judgements of physical attractiveness. Correlations also are the way that we measure validity of measures. Validity is the degree to which a variable measures what it's supposed to. For example, IQ with school performance, with occupational attainment, with income, all of these things are measures of intelligence. Not perfectly. There's a lot that goes into school performance other than just intelligence. A lot that goes with occupational attainment and income other than just intelligence. But if you didn't get a correlation between an IQ test with these things, let's say, gee, I don't think that's a measure of intelligence at all. And, in fact, all of those correlations run about 0.40 to 0.50. So they have some real validity defined in that way. Or we could look at the validity of a paper and pencil test of extroversion, how outgoing someone is with ratings of extroversion of the same people across a number of situations. So we follow them around a party, a classroom, a committee meeting, and we see how outgoing they seem to be. We give them a rating across all those situations and we correlate that with the paper and pencil test of extroversion. I love loud parties or I like nothing better than curling up with a book, and what you find is those correlations tend to run about 0.7. So personality traits can be measured quite reliably. However, and this is to steal a march on the next lesson, if you were to correlate that paper and pencil test of extroversion, that score, with the ratings of extroversion in a single situation, you get a number more like 0.15 or 0.2. So you can't get terribly good prediction about people along personality trait lines by virtue of having only a single score. Well, we could look at Rorschach ratings of paranoia with psychiatric ratings of paranoia. There's a Rorschach test, an ink blot, and so, somebody might say, that's a very scary thing there, their eyes creep me out and maybe that person is paranoid. But the actual correlation between judgments of paranoia by the Rorschach or judgments of anything else by the Rorschach is about zero. The Rorschach test has no validity. And there's a moral to that story, which is that, for decades people were given Rorschach's tests by psychologists and psychiatrists, and there was no validity at all. Millions of dollars were spent and hundreds of thousands of hours of people's time was wasted and wrong conclusions about them were drawn. Because no one ever took the trouble to see how valid these tests were. There are two very important points I want to make about the relationship between validity and reliability. The first point is that there can be no validity if there is no reliability. If your measure gives you a different score every time and they're more or less random, so that Joe gets a high score one measurement and a low score on another. And Tom gets a high score on one and a low score on the other and there's no relationship at all, then you can't have any validity for that measure. There has to be some stability, some degree of getting the same answer twice before you can have any validity for that measure at all. A friend of mine, for example, found out that he has mold in his apartment. The question is, how much mold is there in the apartment? That sounds like a straight forward question, but it turns out that mold varies from moment to moment. I mean, and now there's a lot of mold in the spores in the air, and the corner, and 15 minutes later, there isn't much there. The measures that we have of Mold are not very reliable and therefore, they can't be very valid. A second point is that reliability implies very little about validity. Now, if reliability is zero, there can't be any validity. But at the other extreme, reliability can be absolutely perfect, but there may be no validity. And we have a case study of that in the Rorschach. People learned how to score Rorschach tests from first Dr. Rorschach, started teaching people how to do it, and they taught their students, and their students taught students, and they all were doing it the same way, it's just that there was no relationship to reality of these various measures that they. So we've been talking about correlations. And correlation is one of the main ways that science gets done. Next time, we'll be discussing an extremely important principle of probability, namely the law of large numbers. And that's an important concept to have in knowing whether we can trust our judgment based on the amount of evidence that we have about some matter.